A NEW CHARACTERIZATION OF GEODESIC SPHERES IN THE HYPERBOLIC SPACE.

This paper gives a new characterization of geodesic spheres in the hyperbolic space in terms of a "weighted" higher order mean curvature. Precisely, we show that a compact hypersurface Σn-1 embedded in ℍn with VHk being constant for some k = 1, · · ·, n - 1 is a centered geodesic sphere. Here Hk is the k-th normalized mean curvature of Σ induced from Hn and V = coshr, where r is a hyperbolic distance to a fixed point in ℍn. Moreover, this result can be generalized to a compact hypersurface Σ embedded in Hn with the ratio V (Hk/Hj) ≡ constant, 0 ≤ j < k ≤ n - 1 and Hj not vanishing on Σ. [ABSTRACT FROM AUTHOR]